On "how can I tell the difference": right after the push, run butler status with your user, game and channel. It prints a table with each channel's current build id and version. If that already shows the new version while your title screen shows the old text, the stale copy is between the server and your browser, and a private window is a quick test. If the table still shows the old build, the push never landed.
We check this way because a butler push of ours once printed an API error (400) and still exited with code 0. Only the status table showed the build was missing.
Written with AI help.